Performance Evaluation
Lecture 2: Complex Networks
Giovanni Neglia
INRIA – EPI Maestro
16 December 2013
Configuration model
A family of random graphs with given
degree distribution
Configuration model
A family of random graphs with given
degree distribution
Uniform random matching of stubs
Configuration model
A family of random graphs with given
degree distribution
Uniform random matching of stubs
Back to Navigation:
Random Walks
What can we do in networks without a
geographical structure?
Random walks
1/4 1/4
1/2
1/4
1/2
1/4
Back to Navigation:
Random Walks
How much time is needed in order to reach
a given node?
Random Walks:
stationary distribution
1
pi = å p j
jÎN i k j
ki
ki
pi = N =
2M
åk j
i
i=1
avg time to come back to node i starting from
node i:
1 2M
=
pi
ki
Avg time to reach node i
intuitively ≈Θ(M/ki)
Another justification
Random walk as random edge sampling
Prob. to pick an edge (and a direction) leading to a
kpk
node of degree k is
<k>
Prob. to arrive to a given node of degree k:
kpk
k
=
pk N < k > 2M
Avg. time to arrive to this node 2M/k
…equivalent to a RW where at each step we
sample a configuration model
Distributed navigation
(speed up random walks)
Every node knows its neighbors
{a,b,c,d}
a
b
d
i
c
Distributed navigation
(speed up random walks)
Every node knows its neighbors
If a random walk looking for i arrives in a the
message is directly forwarded to i
{a,b,c,d}
a
b
d
i
c
Distributed navigation
reasoning 1
We discover i when we sample one of the
links of i’s neighbors
æ < k2 > ö
æ
kpk ö
-1÷
Avg # of these links: ki åç (k -1)
÷ = ki ç
è
< k >ø
è <k> ø
k
2
æ
ö
k
<
k
>
i
Prob. to arrive at one of them:
-1÷
ç
2M è < k > ø
a
b
d
i
c
Distributed navigation
reasoning 2
Prob that a node of degree k is neighbor of
node i given that RW arrives to this node from
a node different from i
k-1
æ
ki ö
ki (k -1)
1- ç1÷ »
è 2M ø
2M
Prob that the next edge brings to a node that
is neighbor of node i:
ki (k -1) kpk
ki æ < k 2 > ö
å 2M < k > = 2M çè < k > -1÷ø
k
Distributed navigation
Avg. Hop#
2M
<k>
ki < k 2 > - < k >
2M
Regular graph with degree d:
d(d -1)
2M
ER with <k>:
ki (< k > -1)
æ
a
xma ö
Pareto distribution ç P(k) » a +1 ÷ :
x ø
è
2M (a - 2)(a -1)
If α->2…
»
ki xm - (a - 2)(a -1)
Distributed navigation
Application example:
File search in unstructured P2P networks
through RWs